Topological qubits based on Majorana fermions have the potential to revolutionize the emerging field of quantum computing by making information processing significantly more robust to decoherence. Nanowires (NWs) are a promising medium for hosting these kinds of qubits, though branched NWs are needed to perform qubit manipulations. Here we report gold-free templated growth of III-V NWs by molecular beam epitaxy using an approach that enables patternable and highly regular branched NW arrays on a far greater scale than what has been reported thus far. Our approach relies on the lattice-mismatched growth of InAs on top of defect-free GaAs nanomembranes (NMs) yielding laterally-oriented, low-defect InAs and InGaAs NWs whose shapes are determined by surface and strain energy minimization. By controlling NM width and growth time, we demonstrate the formation of compositionally graded NWs with cross-sections less than 50 nm. Scaling the NWs below 20 nm leads to the formation of homogenous InGaAs NWs which exhibit phase-coherent, quasi-1D quantum transport as shown by magnetoconductance measurements. These results are an important advance towards scalable topological quantum computing.